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Counting dimensions of L-harmonic functions with exponential growth
Geometriae Dedicata ( IF 0.5 ) Pub Date : 2020-03-09 , DOI: 10.1007/s10711-020-00520-y
Xian-Tao Huang

Let $$\Omega \subset {\mathbb {R}}^{n-1}$$ Ω ⊂ R n - 1 be a bounded open set, $$X=\Omega \times {\mathbb {R}}\subseteq {\mathbb {R}}^{n}$$ X = Ω × R ⊆ R n be the infinite strip. Let L be a second order uniformly elliptic operator of divergence form acting on a function $$f\in W_{\text {loc}}^{1,2}(X)$$ f ∈ W loc 1 , 2 ( X ) given by $$Lf=\sum _{i,j=1}^{n}\frac{\partial }{\partial x_{i}}\bigl (a^{ij}(x)\frac{\partial f}{\partial x_{j}}\bigr )$$ L f = ∑ i , j = 1 n ∂ ∂ x i ( a ij ( x ) ∂ f ∂ x j ) . It is natural to consider the solutions of $$Lu=0$$ L u = 0 with boundary value $$u|_{\partial \Omega \times {\mathbb {R}}}=0$$ u | ∂ Ω × R = 0 and exponential growth at most d : $$|u(x',x_{n})|\le {\tilde{C}}e^{d|x_{n}|}$$ | u ( x ′ , x n ) | ≤ C ~ e d | x n | for some $${\tilde{C}}>0$$ C ~ > 0 . Denote by $${\mathcal {A}}_{d}$$ A d the solution space. In (Acta Math Sin (Engl Ser)15:525–534, 1999), Hang and Lin proved that $$\text {dim}{\mathcal {A}}_{d}\le Cd^{n-1}$$ dim A d ≤ C d n - 1 . The power $$n-1$$ n - 1 is sharp, but one may wonder whether there are more precise estimates for the constant C . In this note, we consider some natural subspaces of $${\mathcal {A}}_{d}$$ A d and obtain some estimates of dimensions of these subspaces. Compared with the case $$L=\Delta _{X}$$ L = Δ X , when d is sufficiently large, the estimates obtained in this note are sharp both on the power $$n-1$$ n - 1 and the constant C .

中文翻译:

计算具有指数增长的 L 谐波函数的维数

令 $$\Omega \subset {\mathbb {R}}^{n-1}$$ Ω ⊂ R n - 1 是一个有界开集,$$X=\Omega \times {\mathbb {R}}\ subseteq {\mathbb {R}}^{n}$$ X = Ω × R ⊆ R n 是无限长条。令 L 为作用于函数 $$f\in W_{\text {loc}}^{1,2}(X)$$ f ∈ W loc 1 , 2 ( X ) 的散度形式的二阶一致椭圆算子由 $$Lf=\sum _{i,j=1}^{n}\frac{\partial }{\partial x_{i}}\bigl (a^{ij}(x)\frac{\partial f}{\partial x_{j}}\bigr )$$ L f = ∑ i , j = 1 n ∂ ∂ xi ( a ij ( x ) ∂ f ∂ xj ) 。很自然地考虑 $$Lu=0$$ L u = 0 的解,边界值为 $$u|_{\partial \Omega \times {\mathbb {R}}}=0$$ u | ∂ Ω × R = 0 且最多 d 呈指数增长: $$|u(x',x_{n})|\le {\tilde{C}}e^{d|x_{n}|}$$ | u ( x ′ , xn ) | ≤ C ~ ed | xn | 对于某些 $${\tilde{C}}>0$$ C ~ > 0 。用 $${\mathcal {A}}_{d}$$ A 表示解空间。在 (Acta Math Sin (Engl Ser)15:525–534, 1999) 中,Hang 和 Lin 证明了 $$\text {dim}{\mathcal {A}}_{d}\le Cd^{n-1} $$ dim A d ≤ C dn - 1 。$$n-1$$ n - 1 的幂是尖锐的,但人们可能想知道是否有更精确的常数 C 估计。在这篇笔记中,我们考虑 $${\mathcal {A}}_{d}$$ A d 的一些自然子空间,并获得这些子空间的维度的一些估计。与 $$L=\Delta _{X}$$ L = Δ X 的情况相比,当 d 足够大时,本文中获得的估计在 $$n-1$$n - 1 和常数 C 。但人们可能想知道是否有更精确的常数 C 估计。在本笔记中,我们考虑 $${\mathcal {A}}_{d}$$ A d 的一些自然子空间,并获得这些子空间的一些维度估计。与 $$L=\Delta _{X}$$ L = Δ X 的情况相比,当 d 足够大时,本文中获得的估计在 $$n-1$$n - 1 和常数 C 。但人们可能想知道是否有更精确的常数 C 估计。在这篇笔记中,我们考虑 $${\mathcal {A}}_{d}$$ A d 的一些自然子空间,并获得这些子空间的维度的一些估计。与 $$L=\Delta _{X}$$ L = Δ X 的情况相比,当 d 足够大时,本文中获得的估计在 $$n-1$$n - 1 和常数 C 。
更新日期:2020-03-09
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